General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 117

The positive equilibrium

16.N 1/.N C 4/
.6N 1/ 2 .6N 1/ 2

; 80.N 1/ x D

is in region D

.1/ , whereas no equilibria exist in regions

D

.k/ , k D 2;3;4.Inorder

to study the local stability of the positive fixed point x, we consider the Jacobian

matrix

0

a 1 h .N 1/ i

1

2.N 1/
p 5.N

1 a 1

J .1/

@

a 2 h 1 i 1 a 2 C .N 2/a 2 h

1/x 2

1 i

A

D

p 2
p 3Œx 1 C

p 2
p 3Œx 1 C

.N 2/x 2 .N 2/x 2

(3.16)

C
1246.N 1/>0 given before follows after some calculation. As noticed above,
the equilibrium x undergoes a flip (or period doubling) bifurcation for increasing
N. After the first flip bifurcation, occurring at N ' 13, further period doublings
occur and a route towards chaotic behavior is observed for increasing values of
N. However, it is obvious from the stability conditions in (3.15) that the values of
the two speeds of adjustment also play an important role. Stability of the positive
equilibrium is always ensured for appropriately selected low values of the adjust-
ment speed a 2 . This can also be confirmed by numerical simulations. In Fig. 3.7 we
show a bifurcation diagram obtained with N D 23, where all the other parameters
are chosen as in Fig. 3.6 and with the bifurcation parameter a 2 spanning the whole
range .0;1. For low values of a 2 the equilibrium is stable. For increasing values
of a 2 several sudden transitions between chaotic and periodic behavior characterize
the asymptotic dynamics. Many of these bifurcations are different from the common
bifurcations observed for smooth dynamical systems as the reader might notice. The
reason is that the bifurcations observed here are strongly influenced by the pres-
ence of the lines of non-differentiability. As already stressed in Chap. 2, these can
be often classified as border collision bifurcations, occurring when an equilibrium
point (or a periodic point) of a piecewise differentiable dynamical system crosses
a curve of non-differentiability. Such a contact may produce many kinds of effects
(transition to another cycle of any period or a sudden transition to chaos) depend-
ing on the eigenvalues of the two Jacobian matrices on the two adjacent sides of
the curve of non-differentiability involved in the contact (see for example, Banerjee
et al. (2000b)). Moreover, as we have shown in Chap. 2 (see also Appendix C) the
lines of non-differentiability may represent “folding lines,” and consequently they
have a role similar to that of the critical curves, where the latter are defined as sets
of points where the Jacobian determinant vanishes. In other words, candidates for
the “folding curves” F .i / in the particular example we are considering are:

x. Using the characteristic equation, the stability condition 88N 2

computed at

16
5.N

3x 1
3.N

8

1. The curves of non-differentiability, that is the lines x 2 D

2/ ;
2. The curves of vanishing Jacobian, where the Jacobian matrices in the regions
D

and x 2 D

1/

.k/ , k D 1;:::;4, are respectively J .1/ , given in (3.16),